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Theorem funimass3 5657
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5656 would be the special case of  A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
funimass3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )

Proof of Theorem funimass3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funimass4 5589 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssel 3187 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
3 fvimacnv 5656 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  e.  B  <->  x  e.  ( `' F " B ) ) )
43ex 423 . . . . . 6  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  (
( F `  x
)  e.  B  <->  x  e.  ( `' F " B ) ) ) )
52, 4syl9r 67 . . . . 5  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( x  e.  A  ->  (
( F `  x
)  e.  B  <->  x  e.  ( `' F " B ) ) ) ) )
65imp31 421 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  e.  B  <->  x  e.  ( `' F " B ) ) )
76ralbidva 2572 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  e.  B  <->  A. x  e.  A  x  e.  ( `' F " B ) ) )
81, 7bitrd 244 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  x  e.  ( `' F " B ) ) )
9 dfss3 3183 . 2  |-  ( A 
C_  ( `' F " B )  <->  A. x  e.  A  x  e.  ( `' F " B ) )
108, 9syl6bbr 254 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556    C_ wss 3165   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  funimass5  5658  funconstss  5659  fvimacnvALT  5660  fimacnv  5673  r0weon  7656  iscnp3  16990  cnpnei  17009  cnclsi  17017  cncls  17019  cncnp  17025  1stccnp  17204  txcnpi  17318  xkoco2cn  17368  xkococnlem  17369  basqtop  17418  kqnrmlem1  17450  kqnrmlem2  17451  reghmph  17500  nrmhmph  17501  elfm3  17661  rnelfm  17664  symgtgp  17800  tgpconcompeqg  17810  eltsms  17831  plyco0  19590  plyeq0  19609  xrlimcnp  20279  rinvf1o  23054  xppreima  23226  cvmliftmolem1  23827  cvmlift2lem9  23857  cvmlift3lem6  23870  cmptdst  25671  flfnei2  25680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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